phenomenon 30.32 Poisson contraction

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:972 · p. 1018

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on phenomenon 30.28 Hooke's law ¶
- depends_on definition 30.24 The elasticity tensor ¶
  - depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
    - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
    - depends_on equation 30.2 eq:elast-displacement ¶
  - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
    - depends_on definition 30.17 Traction ¶
      - depends_on definition 7.126 Line and surface integrals ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
    - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 13.4 Orthogonal coordinate transformation ¶
    - proves proof ch:13-continuum-elasticity@proof-7 ¶
- depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
  - depends_on definition 13.5 Tensor under orthogonal transformations ¶ ↺
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - proves proof ch:03-linear-algebra-representations@proof-8 ¶
    - depends_on lemma 5.132 Parity constraint ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on equation 5.162 eq:rep-special-orthogonal ¶
      - proves proof ch:03-linear-algebra-representations@proof-57 ¶
    - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - proves proof ch:03-linear-algebra-representations@proof-56 ¶
    - proves proof ch:03-linear-algebra-representations@proof-58 ¶
  - proves proof ch:13-continuum-elasticity@prooflink-1 ¶
- proves proof ch:13-continuum-elasticity@proof-12 ¶
depends_on proposition 30.30 Stability bounds ¶
- depends_on equation 30.23 eq:elast-energy-density ¶
- depends_on theorem 5.116 Sylvester's law of inertia ¶
  - depends_on definition 5.110 Bilinear map ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-53 ¶
- proves proof ch:13-continuum-elasticity@proof-13 ¶
proves proof ch:13-continuum-elasticity@proof-14 ¶

Supports

depends_on proposition 30.33 Conversion among the moduli ¶
- depends_on definition 30.63 Flexural rigidity ¶
  - depends_on proposition A.684 The plate energy ¶
    - depends_on lemma A.685 The Gaussian-curvature term is a null Lagrangian ¶
      - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶
    - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on remark A.690 Poisson's count, and why it cannot be right ¶
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
  - depends_on proposition A.682 The plane-stress law ¶
    - depends_on definition A.689 Edge resultants ¶
      - depends_on remark A.690 Poisson's count, and why it cannot be right ¶ ↺
      - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶ ↺
    - depends_on proposition A.684 The plate energy ¶ ↺
    - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶
  - depends_on proposition 30.66 Chladni scaling ¶
    - depends_on phenomenon 30.65 Chladni figures ¶
  - depends_on remark A.704 $E^{*}$ in two roles, and one trap ¶
  - depends_on remark A.686 Why the operator comes out as $\nabla^{4}$ ¶ ↺
  - depends_on remark A.683 Where the factor $1-\nu^{2}$ comes from ¶ ↺
  - depends_on remark 30.68 Shells carry load in a different way ¶
  - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on phenomenon 30.65 Chladni figures ¶ ↺
    - depends_on proposition 30.66 Chladni scaling ¶ ↺
    - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
- depends_on example 30.35 From single crystals to the isotropic aggregate ¶
- depends_on proposition A.682 The plane-stress law ¶ ↺
- depends_on proposition 30.44 The speed ratio is fixed by Poisson's ratio ¶
  - depends_on example 30.45 Elastic waves in steel, and the distance to a source ¶
- depends_on remark 30.36 How the moduli are actually measured ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
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Edges

type	direction	node	provenance	where
`cites`	→	Mémoire sur l'équilibre et le mouvement des corps élastiques	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:976
`cites`	→	Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:985
`cites`	→	A Course of Lectures on Natural Philosophy and the Mechanical Arts	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:978
`depends_on`	→	Hooke's law	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:986
`depends_on`	→	Stability bounds	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:986
`depends_on`	←	Conversion among the moduli	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1036
`proves`	←	ch:13-continuum-elasticity@proof-14	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:989